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First order Griewank function
In mathematics, the Griewank function is often used in testing of optimization. It is defined as follows:[1]
![{\displaystyle 1+{\frac {1}{4000}}\sum _{i=1}^{n}x_{i}^{2}-\prod _{i=1}^{n}\cos \left({\frac {x_{i}}{\sqrt {i}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/636a078b20d38802f4236ef1839eb3f521cbfa2f)
The following paragraphs display the special cases of first, second and third order
Griewank function, and their plots.
First-order Griewank function[edit]
![{\displaystyle g:=1+(1/4000)\cdot x_{1}^{2}-\cos(x_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec24e0967d4fe5e41e1f0339642f945edc5e9260)
The first order Griewank function has multiple maxima and minima.[2]
Let the derivative of Griewank function be zero:
![{\displaystyle {\frac {1}{2000}}\cdot x_{1}+\sin(x_{1})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1a279835e519e43b5adba8ebf1f29fc1b17bbd)
Find its roots in the interval [−100..100] by means of numerical method,
In the interval [−10000,10000], the Griewank function has 6365 critical points.
Second-order Griewank function[edit]
2nd order Griewank function 3D plot
2nd-order Griewank function contour plot
![{\displaystyle 1+{\frac {1}{4000}}x_{1}^{2}+{\frac {1}{4000}}x_{2}^{2}-\cos(x_{1})\cos \left({\frac {1}{2}}x_{2}{\sqrt {2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c915e0fce8fc80dc4fedce5d43b069e2b1de9d)
Third order Griewank function[edit]
Third-order Griewank function Maple animation
![{\displaystyle \left\{1+{\frac {1}{4000}}\,x_{1}^{2}+{\frac {1}{4000}}\,x_{2}^{2}+{\frac {1}{4000}}\,{x_{3}}^{2}-\cos(x_{1})\cos \left({\frac {1}{2}}x_{2}{\sqrt {2}}\right)\cos \left({\frac {1}{3}}x_{3}{\sqrt {3}}\right)\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73d55db3701007e914d312a286f4d16e9736b174)
References[edit]
- ^ Griewank, A. O. "Generalized Descent for Global Optimization." J. Opt. Th. Appl. 34, 11–39, 1981
- ^ Locatelli, M. "A Note on the Griewank Test Function." J. Global Opt. 25, 169–174, 2003